For $N_\lambda$ denoting the number of eigenvalues less than $\lambda$, Weyl's law gives the asymptotics of $N_\lambda$ as $\lambda$ tends to infinity. The usual approach to establish this asymptotics in the continuum divides space into boxes. This fails for a graph, since cutting the graph is not a small perturbation.   
An analogue of Weyl asymptotics for graph Laplacians that satisfy a strong isoperimetric inequality is given in <A HREF="https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-GraphTh/Keller/KellerLenzWojciechowski_GraphsAndDiscreteDirichletSpaces_wu_version.pdf">Graphs and Discrete Dirichlet Spaces</A> by Keller, Lenz, and Wojciechowski, page 439. See also <A HREF="https://hal.science/hal-01010730/document">Essential spectrum and Weyl asymptotics for discrete Laplacians</A> by Bonnefont and Golenia.

The above refers to the eigenvalues. For the eigenvectors, the localisation phenomenon has been studied in <A HREF="https://www.nature.com/articles/s41598-017-01010-0">Localization of Laplacian eigenvectors on random networks</A>, but there is no "rigorous" theory.